Evaluate\sin ^( - 1) ( \sin( \binom{3\pi}{4} ) )Show your reasoning.

Question

Evaluate
`\sin ^( - 1) ( \sin( \binom{3\pi}{4} ) )`Show your reasoning.

Answer 1

`\sin ^(-1)(\sin ((3\pi)/(4)))`

We will first resolve the expression in the parenthesis

3/4pi can be converted to degrees:

since 1 pi = 180 degrees

3/4 pi = x degrees

`\begin{gathered} x=\frac{3\pi\text{ x 180}}{4\text{ }\pi} \\ =135^0 \end{gathered}`

sin (135) = 0.707 or

`\sin \text{ 135 = }\frac{\sqrt[]{2}}{2}`

We can then find

`\begin{gathered} \sin ^(-1)(\frac{\sqrt[]{2}}{2}) \\ =(3\pi)/(4) \end{gathered}`

So

`\sin ^(-1)(\sin ((3\pi)/(4)))\text{ = }(3\pi)/(4)`